Yes, math enrichment week (more about it here) is finally over! I personally believe that it was a massive success, definitely because everything went well as planned, but also because it opened up new opportunities for me. One of them is a new intern place I got at INRIA, the French national institute of computational sciences. Basically, computational sciences is a wide all-encompassing field that ranges from hardware science to mathematics, which my employer (I don't know what to call him?) Dr Julien Salomon specializes in.
So how I got to know him is like this. One of the projects that we were initiating was a math concert lecture, which is a series of lectures by two professors followed by a musical entr'acte. Dr Salomon was one of the two lecturers, and was invited by my math teacher Mr Munier. The other professor was a professor in applied mathematics as well, but in physics. Although I love physics as well, I was personally more fascinated by Dr Salomon's research (and my intended major of study is closer to Dr Salomon as well), so I saw this as a chance to earn an internship place. As the math enrichment week was successfully finished and I had done lots of work, I asked Mr Munier if I could intern at Dr Salomon's place. I had personally been very engaged in Mr Munier's classes and became very close while preparing for the action week, and he recommended me as "the best student he had seen so far" (!) to Dr Salomon!! Dr Salomon said that then we can begin work right away, and thus I earned my internship position without having to go through the uncertain process of sending hundreds of resumes and emails to professors who I do not know. (Someone that I know literally sent 200+ emails yet never got a reply.)
I was then allowed to begin work, and today was the first day, where I did a meeting with him to learn about the basics of his area of research--optimal transport. As the name itself is quite self-explanatory, optimal transport is basically the study of transporting masses from a series of supply points (e.g. mines, factories, etc) to several demand points (e.g. train stations, shops, marts, etc).
He also introduced a special case of the optimal transport problem, which involves concave costs (graph in the figure below) and is an one-dimensional case. (all points are colinear)
I hope that these diagrams make it a bit clear
Dr Salomon's research was on deducing the global optimal transport plan from evaluating a function called "indicators" only on local structures. So, the theorem is essentially like this. If the cost of transport on the right is greater or equal than the cost of transport in the left, an optimal transport plan must match the two points in the bottom figure.
So, the importance of such research seems very clear! Computing every single transport plan would not only be a waste of time, but it may also be simply unfeasible in real-life contexts that involve thousands of supplies and demands. This research also seemed very fascinating. I could already think of applications of this theorem or possible extensions. What about 2 dimensions? What about graphs? How could this be applied to convex costs? Could this help renewing the public transportation network in Jeju, where the buses that come once in 30 minutes is a massive bummer?
So, my first task is to complete proof for a special case within the paper that Dr Salomon has written. I also received the formal proof of the theorem, which looks, honestly, very intimidating, but I will nevertheless try my best to not let my math teacher down. One very annoying aspect is that I will have to stop work in a week because of my annual exams, which last a month. But after that, I'll try to keep this blog updated with new works. A paper will soon be published on the view all research page!